Integrand size = 18, antiderivative size = 146 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=a^4 x+3 a^3 b x^{4/3}+\frac {6}{5} a^2 \left (3 b^2+2 a c\right ) x^{5/3}+2 a b \left (b^2+3 a c\right ) x^2+\frac {3}{7} \left (b^4+12 a b^2 c+6 a^2 c^2\right ) x^{7/3}+\frac {3}{2} b c \left (b^2+3 a c\right ) x^{8/3}+\frac {2}{3} c^2 \left (3 b^2+2 a c\right ) x^3+\frac {6}{5} b c^3 x^{10/3}+\frac {3}{11} c^4 x^{11/3} \] Output:
a^4*x+3*a^3*b*x^(4/3)+6/5*a^2*(2*a*c+3*b^2)*x^(5/3)+2*a*b*(3*a*c+b^2)*x^2+ 3/7*(6*a^2*c^2+12*a*b^2*c+b^4)*x^(7/3)+3/2*b*c*(3*a*c+b^2)*x^(8/3)+2/3*c^2 *(2*a*c+3*b^2)*x^3+6/5*b*c^3*x^(10/3)+3/11*c^4*x^(11/3)
Time = 0.12 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.12 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=\frac {2310 a^4 x+6930 a^3 b x^{4/3}+8316 a^2 b^2 x^{5/3}+5544 a^3 c x^{5/3}+4620 a b^3 x^2+13860 a^2 b c x^2+990 b^4 x^{7/3}+11880 a b^2 c x^{7/3}+5940 a^2 c^2 x^{7/3}+3465 b^3 c x^{8/3}+10395 a b c^2 x^{8/3}+4620 b^2 c^2 x^3+3080 a c^3 x^3+2772 b c^3 x^{10/3}+630 c^4 x^{11/3}}{2310} \] Input:
Integrate[(a + b*x^(1/3) + c*x^(2/3))^4,x]
Output:
(2310*a^4*x + 6930*a^3*b*x^(4/3) + 8316*a^2*b^2*x^(5/3) + 5544*a^3*c*x^(5/ 3) + 4620*a*b^3*x^2 + 13860*a^2*b*c*x^2 + 990*b^4*x^(7/3) + 11880*a*b^2*c* x^(7/3) + 5940*a^2*c^2*x^(7/3) + 3465*b^3*c*x^(8/3) + 10395*a*b*c^2*x^(8/3 ) + 4620*b^2*c^2*x^3 + 3080*a*c^3*x^3 + 2772*b*c^3*x^(10/3) + 630*c^4*x^(1 1/3))/2310
Time = 0.33 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1680, 1140, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx\) |
\(\Big \downarrow \) 1680 |
\(\displaystyle 3 \int \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^4 x^{2/3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle 3 \int \left (x^{2/3} a^4+4 b x a^3+2 \left (3 b^2+2 a c\right ) x^{4/3} a^2+4 b \left (b^2+3 a c\right ) x^{5/3} a+c^4 x^{10/3}+4 b c^3 x^3+2 c^2 \left (3 b^2+2 a c\right ) x^{8/3}+4 b c \left (b^2+3 a c\right ) x^{7/3}+\left (b^4+12 a c b^2+6 a^2 c^2\right ) x^2\right )d\sqrt [3]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {a^4 x}{3}+a^3 b x^{4/3}+\frac {2}{5} a^2 x^{5/3} \left (2 a c+3 b^2\right )+\frac {1}{7} x^{7/3} \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac {2}{9} c^2 x^3 \left (2 a c+3 b^2\right )+\frac {1}{2} b c x^{8/3} \left (3 a c+b^2\right )+\frac {2}{3} a b x^2 \left (3 a c+b^2\right )+\frac {2}{5} b c^3 x^{10/3}+\frac {1}{11} c^4 x^{11/3}\right )\) |
Input:
Int[(a + b*x^(1/3) + c*x^(2/3))^4,x]
Output:
3*((a^4*x)/3 + a^3*b*x^(4/3) + (2*a^2*(3*b^2 + 2*a*c)*x^(5/3))/5 + (2*a*b* (b^2 + 3*a*c)*x^2)/3 + ((b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^(7/3))/7 + (b*c*( b^2 + 3*a*c)*x^(8/3))/2 + (2*c^2*(3*b^2 + 2*a*c)*x^3)/9 + (2*b*c^3*x^(10/3 ))/5 + (c^4*x^(11/3))/11)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k* n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && Fr actionQ[n]
Time = 5.40 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.96
method | result | size |
default | \(a^{4} x +\frac {3 b^{4} x^{\frac {7}{3}}}{7}+\frac {3 c^{4} x^{\frac {11}{3}}}{11}+2 b^{3} a \,x^{2}+\frac {4 a \,c^{3} x^{3}}{3}+\frac {18 a^{2} b^{2} x^{\frac {5}{3}}}{5}+\frac {18 a^{2} c^{2} x^{\frac {7}{3}}}{7}+3 a^{3} b \,x^{\frac {4}{3}}+\frac {12 a^{3} c \,x^{\frac {5}{3}}}{5}+\frac {6 b \,c^{3} x^{\frac {10}{3}}}{5}+2 b^{2} c^{2} x^{3}+\frac {3 b^{3} c \,x^{\frac {8}{3}}}{2}+\frac {9 b a \,c^{2} x^{\frac {8}{3}}}{2}+\frac {36 a \,b^{2} c \,x^{\frac {7}{3}}}{7}+6 a^{2} b c \,x^{2}\) | \(140\) |
derivativedivides | \(\frac {3 c^{4} x^{\frac {11}{3}}}{11}+\frac {6 b \,c^{3} x^{\frac {10}{3}}}{5}+\frac {\left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right ) x^{3}}{3}+\frac {3 \left (4 b a \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) x^{\frac {8}{3}}}{8}+\frac {3 \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) x^{\frac {7}{3}}}{7}+\frac {\left (4 a^{2} b c +4 b a \left (2 a c +b^{2}\right )\right ) x^{2}}{2}+\frac {3 \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right ) x^{\frac {5}{3}}}{5}+3 a^{3} b \,x^{\frac {4}{3}}+a^{4} x\) | \(168\) |
trager | \(\frac {\left (4 a \,c^{3} x^{2}+6 b^{2} c^{2} x^{2}+18 a^{2} b c x +6 a \,b^{3} x +4 a \,c^{3} x +6 b^{2} x \,c^{2}+3 a^{4}+18 a^{2} b c +6 b^{3} a +4 a \,c^{3}+6 b^{2} c^{2}\right ) \left (x -1\right )}{3}+\frac {3 x^{\frac {4}{3}} \left (14 b \,x^{2} c^{3}+30 a^{2} c^{2} x +60 a \,b^{2} x c +5 b^{4} x +35 a^{3} b \right )}{35}+\frac {3 x^{\frac {5}{3}} \left (10 c^{4} x^{2}+165 a b x \,c^{2}+55 b^{3} x c +88 a^{3} c +132 a^{2} b^{2}\right )}{110}\) | \(178\) |
orering | \(\text {Expression too large to display}\) | \(2216\) |
Input:
int((a+b*x^(1/3)+c*x^(2/3))^4,x,method=_RETURNVERBOSE)
Output:
a^4*x+3/7*b^4*x^(7/3)+3/11*c^4*x^(11/3)+2*b^3*a*x^2+4/3*a*c^3*x^3+18/5*a^2 *b^2*x^(5/3)+18/7*a^2*c^2*x^(7/3)+3*a^3*b*x^(4/3)+12/5*a^3*c*x^(5/3)+6/5*b *c^3*x^(10/3)+2*b^2*c^2*x^3+3/2*b^3*c*x^(8/3)+9/2*b*a*c^2*x^(8/3)+36/7*a*b ^2*c*x^(7/3)+6*a^2*b*c*x^2
Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.96 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=a^{4} x + \frac {2}{3} \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{3} + 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{2} + \frac {3}{110} \, {\left (10 \, c^{4} x^{3} + 55 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{2} + 44 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x\right )} x^{\frac {2}{3}} + \frac {3}{35} \, {\left (14 \, b c^{3} x^{3} + 35 \, a^{3} b x + 5 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{2}\right )} x^{\frac {1}{3}} \] Input:
integrate((a+b*x^(1/3)+c*x^(2/3))^4,x, algorithm="fricas")
Output:
a^4*x + 2/3*(3*b^2*c^2 + 2*a*c^3)*x^3 + 2*(a*b^3 + 3*a^2*b*c)*x^2 + 3/110* (10*c^4*x^3 + 55*(b^3*c + 3*a*b*c^2)*x^2 + 44*(3*a^2*b^2 + 2*a^3*c)*x)*x^( 2/3) + 3/35*(14*b*c^3*x^3 + 35*a^3*b*x + 5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)* x^2)*x^(1/3)
Time = 0.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.08 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=a^{4} x + 3 a^{3} b x^{\frac {4}{3}} + \frac {6 b c^{3} x^{\frac {10}{3}}}{5} + \frac {3 c^{4} x^{\frac {11}{3}}}{11} + \frac {3 x^{\frac {8}{3}} \cdot \left (12 a b c^{2} + 4 b^{3} c\right )}{8} + \frac {3 x^{\frac {7}{3}} \cdot \left (6 a^{2} c^{2} + 12 a b^{2} c + b^{4}\right )}{7} + \frac {3 x^{\frac {5}{3}} \cdot \left (4 a^{3} c + 6 a^{2} b^{2}\right )}{5} + \frac {x^{3} \cdot \left (4 a c^{3} + 6 b^{2} c^{2}\right )}{3} + \frac {x^{2} \cdot \left (12 a^{2} b c + 4 a b^{3}\right )}{2} \] Input:
integrate((a+b*x**(1/3)+c*x**(2/3))**4,x)
Output:
a**4*x + 3*a**3*b*x**(4/3) + 6*b*c**3*x**(10/3)/5 + 3*c**4*x**(11/3)/11 + 3*x**(8/3)*(12*a*b*c**2 + 4*b**3*c)/8 + 3*x**(7/3)*(6*a**2*c**2 + 12*a*b** 2*c + b**4)/7 + 3*x**(5/3)*(4*a**3*c + 6*a**2*b**2)/5 + x**3*(4*a*c**3 + 6 *b**2*c**2)/3 + x**2*(12*a**2*b*c + 4*a*b**3)/2
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=\frac {3}{11} \, c^{4} x^{\frac {11}{3}} + \frac {6}{5} \, b c^{3} x^{\frac {10}{3}} + 2 \, b^{2} c^{2} x^{3} + \frac {3}{2} \, b^{3} c x^{\frac {8}{3}} + \frac {3}{7} \, b^{4} x^{\frac {7}{3}} + a^{4} x + \frac {3}{5} \, {\left (4 \, c x^{\frac {5}{3}} + 5 \, b x^{\frac {4}{3}}\right )} a^{3} + \frac {6}{35} \, {\left (15 \, c^{2} x^{\frac {7}{3}} + 35 \, b c x^{2} + 21 \, b^{2} x^{\frac {5}{3}}\right )} a^{2} + \frac {1}{42} \, {\left (56 \, c^{3} x^{3} + 189 \, b c^{2} x^{\frac {8}{3}} + 216 \, b^{2} c x^{\frac {7}{3}} + 84 \, b^{3} x^{2}\right )} a \] Input:
integrate((a+b*x^(1/3)+c*x^(2/3))^4,x, algorithm="maxima")
Output:
3/11*c^4*x^(11/3) + 6/5*b*c^3*x^(10/3) + 2*b^2*c^2*x^3 + 3/2*b^3*c*x^(8/3) + 3/7*b^4*x^(7/3) + a^4*x + 3/5*(4*c*x^(5/3) + 5*b*x^(4/3))*a^3 + 6/35*(1 5*c^2*x^(7/3) + 35*b*c*x^2 + 21*b^2*x^(5/3))*a^2 + 1/42*(56*c^3*x^3 + 189* b*c^2*x^(8/3) + 216*b^2*c*x^(7/3) + 84*b^3*x^2)*a
Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.95 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=\frac {3}{11} \, c^{4} x^{\frac {11}{3}} + \frac {6}{5} \, b c^{3} x^{\frac {10}{3}} + 2 \, b^{2} c^{2} x^{3} + \frac {4}{3} \, a c^{3} x^{3} + \frac {3}{2} \, b^{3} c x^{\frac {8}{3}} + \frac {9}{2} \, a b c^{2} x^{\frac {8}{3}} + \frac {3}{7} \, b^{4} x^{\frac {7}{3}} + \frac {36}{7} \, a b^{2} c x^{\frac {7}{3}} + \frac {18}{7} \, a^{2} c^{2} x^{\frac {7}{3}} + 2 \, a b^{3} x^{2} + 6 \, a^{2} b c x^{2} + \frac {18}{5} \, a^{2} b^{2} x^{\frac {5}{3}} + \frac {12}{5} \, a^{3} c x^{\frac {5}{3}} + 3 \, a^{3} b x^{\frac {4}{3}} + a^{4} x \] Input:
integrate((a+b*x^(1/3)+c*x^(2/3))^4,x, algorithm="giac")
Output:
3/11*c^4*x^(11/3) + 6/5*b*c^3*x^(10/3) + 2*b^2*c^2*x^3 + 4/3*a*c^3*x^3 + 3 /2*b^3*c*x^(8/3) + 9/2*a*b*c^2*x^(8/3) + 3/7*b^4*x^(7/3) + 36/7*a*b^2*c*x^ (7/3) + 18/7*a^2*c^2*x^(7/3) + 2*a*b^3*x^2 + 6*a^2*b*c*x^2 + 18/5*a^2*b^2* x^(5/3) + 12/5*a^3*c*x^(5/3) + 3*a^3*b*x^(4/3) + a^4*x
Time = 18.91 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.86 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=x^{7/3}\,\left (\frac {18\,a^2\,c^2}{7}+\frac {36\,a\,b^2\,c}{7}+\frac {3\,b^4}{7}\right )+a^4\,x+\frac {3\,c^4\,x^{11/3}}{11}+x^{5/3}\,\left (\frac {12\,c\,a^3}{5}+\frac {18\,a^2\,b^2}{5}\right )+x^3\,\left (2\,b^2\,c^2+\frac {4\,a\,c^3}{3}\right )+3\,a^3\,b\,x^{4/3}+\frac {6\,b\,c^3\,x^{10/3}}{5}+2\,a\,b\,x^2\,\left (b^2+3\,a\,c\right )+\frac {3\,b\,c\,x^{8/3}\,\left (b^2+3\,a\,c\right )}{2} \] Input:
int((a + b*x^(1/3) + c*x^(2/3))^4,x)
Output:
x^(7/3)*((3*b^4)/7 + (18*a^2*c^2)/7 + (36*a*b^2*c)/7) + a^4*x + (3*c^4*x^( 11/3))/11 + x^(5/3)*((12*a^3*c)/5 + (18*a^2*b^2)/5) + x^3*((4*a*c^3)/3 + 2 *b^2*c^2) + 3*a^3*b*x^(4/3) + (6*b*c^3*x^(10/3))/5 + 2*a*b*x^2*(3*a*c + b^ 2) + (3*b*c*x^(8/3)*(3*a*c + b^2))/2
Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.95 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^4 \, dx=\frac {x \left (5544 x^{\frac {2}{3}} a^{3} c +8316 x^{\frac {2}{3}} a^{2} b^{2}+10395 x^{\frac {5}{3}} a b \,c^{2}+3465 x^{\frac {5}{3}} b^{3} c +630 x^{\frac {8}{3}} c^{4}+6930 x^{\frac {1}{3}} a^{3} b +5940 x^{\frac {4}{3}} a^{2} c^{2}+11880 x^{\frac {4}{3}} a \,b^{2} c +990 x^{\frac {4}{3}} b^{4}+2772 x^{\frac {7}{3}} b \,c^{3}+2310 a^{4}+13860 a^{2} b c x +4620 a \,b^{3} x +3080 a \,c^{3} x^{2}+4620 b^{2} c^{2} x^{2}\right )}{2310} \] Input:
int((a+b*x^(1/3)+c*x^(2/3))^4,x)
Output:
(x*(5544*x**(2/3)*a**3*c + 8316*x**(2/3)*a**2*b**2 + 10395*x**(2/3)*a*b*c* *2*x + 3465*x**(2/3)*b**3*c*x + 630*x**(2/3)*c**4*x**2 + 6930*x**(1/3)*a** 3*b + 5940*x**(1/3)*a**2*c**2*x + 11880*x**(1/3)*a*b**2*c*x + 990*x**(1/3) *b**4*x + 2772*x**(1/3)*b*c**3*x**2 + 2310*a**4 + 13860*a**2*b*c*x + 4620* a*b**3*x + 3080*a*c**3*x**2 + 4620*b**2*c**2*x**2))/2310